Alternative proof for $\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$

Question

Can I use only strong induction in order to prove $$\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$$ Is there any other option?

Answer 1

if $n=3a+4b$ then $n+12k$ is also expressible in this form, so it suffices to show the proposition is true for $n \in [6,17]$

Answer 2

It suffices to discuss the below three cases (below, k ∈N and 1<=k)

1. $n=3k = 3*a$
2. $n=3k+1 = 3*(k-1) + 4 = 3*a + 4*1$
3. $n=3k+2 = 3*(k-2) + 8 = 3*(k-2) + 8 = 3*a + 4*2$

Answer 3

We show that "usual" induction, i.e. going from $n$ to $n+1$, can be made to work here. Assume that $n \ge 6$ and we have a representation $n=3a+4b.$ Then either $a>0$ or not. If $a>0$ we can use one fewer $3$ and one more $4$ and arrive at $n+1.$ On the other hand if $a=0$ we have $n$ as all 4's, and there are at least two 4's involved since $n \ge 6.$ In this case we may decrease by two 4's and increase by three 3's and arrive at $n+1.$

Answer 4

More generally, if $n = pa+qb$ for positive $a, b, p, q$, then $n+2pq =p(a+q)+q(b+p) $.

Therefore, if $n$ can be represented in the form $pa+qb$ for $2pq$ consecutive values of $n$, it can be represented in that form for all larger values of $n$.